We have a quadruple adjunction
\begin{align*} \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(\pi _{0}(\mathcal{C}),X) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},X_{\mathsf{disc}}),\\ \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(X_{\mathsf{disc}},\mathcal{C}) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(X,\operatorname {\mathrm{Obj}}(\mathcal{C})),\\ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(\operatorname {\mathrm{Obj}}(\mathcal{C}),X) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},X_{\mathsf{indisc}}), \end{align*}
natural in $\mathcal{C}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$ and $X\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, where
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The functor
\[ \pi _{0}\colon \mathsf{Cats}\to \mathsf{Sets}, \]the connected components functor, is the functor sending a category to its set of connected components of Definition 11.3.2.2.1.
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The functor
\[ (-)_{\mathsf{disc}}\colon \mathsf{Sets}\to \mathsf{Cats}, \]the discrete category functor, is the functor sending a set to its associated discrete category of Item 1.
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The functor
\[ \operatorname {\mathrm{Obj}}\colon \mathsf{Cats}\to \mathsf{Sets}, \]the object functor, is the functor sending a category to its set of objects.
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The functor
\[ (-)_{\mathsf{indisc}}\colon \mathsf{Sets}\to \mathsf{Cats}, \]the indiscrete category functor, is the functor sending a set to its associated indiscrete category of Item 1.
